Μαθηματικός Εκφυλισμός
Εκφυλισμός Degeneracy - Μία διαδικασία. Ετυμολογία Η ονομασία "Εκφυλισμός" σχετίζεται ετυμολογικά με την λέξη "φυλή". Εισαγωγή A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class * Γραφηματικός Εκφυλισμός (graph theory), a measure of the sparseness of a graph * Εκφυλισμένη Μορφή (Degenerate form), bilinear form ƒ(x,y) on a vector space V'' is one such that the map from to (the dual space of ) given by is not an isomorphism * Εκφυλισμένη Κατανομή ( = Degenerate distribution), the probability distribution of a random variable which only takes a single value * Εκφυλισμένος Κώνος ( = Degenerate conic), a conic (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve * Εκφυλισμένη Διάσταση ( = Degenerate dimension), a dimension key in the fact table that does not have its own dimension table, because all the interesting attributes have been placed in analytic dimensions. The term "degenerate dimension" was originated by Ralph Kimball Ανάλυση In mathematics, a '''degenerate case' is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case. A degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation. In geometry Conic section A degenerate conic is a conic section (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve. * A point is a degenerate circle, namely one with radius 0. * The line is a degenerate case of a parabola if the parabola resides on a tangent plane. * A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. * An ellipse can also degenerate into a single point. * A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common asymptotes. Triangle * A degenerate triangle has collinear vertices and zero area, and thus coincides with a segment covered twice. Rectangle * A segment is a degenerate case of a rectangle, if this has a side of length 0. * For any non-empty subset S \subseteq \{1, 2, \ldots, n\} , there is a bounded, axis-aligned degenerate rectangle : R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\} where \mathbf{x} \triangleq x_2, \ldots, x_n and a_i, b_i, c_i are constant (with a_i \leq b_i for all i ). The number of degenerate sides of R is the number of elements of the subset S . Thus, there may be as few as one degenerate "side" or as many as n (in which case R reduces to a singleton point). Polyhedron * A polyhedron such as a tetrahedron or other pyramid is degenerate if all of its vertices lie in the same plane, giving it zero volume. Standard torus * A sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it. Sphere * When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a point. Elsewhere * A set containing a single point is a degenerate continuum. * A random variable which can only take one value has a degenerate distribution; its probability density is the Dirac Delta function at one point. * Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an ''n''th degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiple coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector. * In quantum mechanics any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels. Usually any such degeneracy indicates some underlying symmetry in the system. See also Υποσημειώσεις Εσωτερική Αρθρογραφία * Εκφυλισμένη Διάσταση * Degenerate form * Trivial (mathematics) * Pathological (mathematics) * Vacuous truth * Κοινωνικός Εκφυλισμός (στην Κοινωνιολογία) * Φιλοσοφικός Εκφυλισμός (στην Φιλοσοφία) * Κβαντικός Εκφυλισμός (Degenerate energy levels) * Βιολογικός Εκφυλισμός (στην Βιολογία) * Μαθηματικός Εκφυλισμός (στα Μαθηματικά) Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Κατηγορία:Μαθηματικά